Edge (geometry)

In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment on the boundary, and is often called a side. (Thus a segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal.) In a polyhedron or more generally a polytope, an edge is a line segment where two two-dimensional faces meet.

Relation to edges in graphs

In graph theory, an edge is an abstract object connecting two graph vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, a graph whose vertices are the geometric vertices of the polyhedron and whose edges correspond to the geometric edges. Conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitz's theorem as being exactly the 3-vertex-connected planar graphs.

Geometric group action

In mathematics, specifically geometric group theory, a geometric group action is a certain type of action of a discrete group on a metric space.

Definition

In geometric group theory, a geometry is any proper, geodesic metric space. An action of a finitely-generated group G on a geometry X is geometric if it satisfies the following conditions:

Uniqueness

If a group G acts geometrically upon two geometries X and Y, then X and Y are quasi-isometric. Since any group acts geometrically on its own Cayley graph, any space on which G acts geometrically is quasi-isometric to the Cayley graph of G.

Examples

Cannon's conjecture states that any hyperbolic group with a 2-sphere at infinity acts geometrically on hyperbolic 3-space.

References

Geometry (disambiguation)

Geometry is a branch of mathematics dealing with spatial relationships.

Geometry or geometric may also refer to:

Pseudoscientific metrology

Some approaches in the branch of historic metrology are highly speculative and can be qualified as pseudoscience.

Origins

In 1637 John Greaves, professor of geometry at Gresham College, made his first of several studies in Egypt and Italy, making numerous measurements of buildings and monuments, including the Great Pyramid. These activities fuelled many centuries of interest in metrology of the ancient cultures by the likes of Isaac Newton and the French Academy.

The pendulum

The first known description and practical use of a physical pendulum is by Galileo Galilei, however, Flinders Petrie, a disciple of Charles Piazzi Smyth, is of the opinion that it was used earlier by the ancient Egyptians. In a 1933 letter to Nature, Petrie wrote:

If we take the natural standard of one day divided by 10⁵, the pendulum would be 29.157 inches at lat 30 degrees. Now this is exactly the basis of Egyptian land measures, most precisely known through the diagonal of that squared, being the Egyptian double cubit. The value for this cubit is 20.617 inches, while the best examples in stone are 20.620±0.005inches.

Characters of the Final Fantasy IV series

Final Fantasy IV, a role-playing video game released by Square in 1991, revolves Cecil Harvey, a knight of Baron who embarks on a quest to defeat Golbez, a man that is controlling the king of Baron. During Cecil's quest, he is joined by his childhood friends Kain Highwind and Rosa Farrell, as well as other warriors from around the world who also seek to stop Golbez. The visuals of the characters were designed by Yoshitaka Amano. After its initial release, Final Fantasy IV was later ported to multiple consoles. In 2007, Square Enix released an enhanced remake for the Nintendo DS that added voice acting to both the Japanese and English versions.

The game's 2008 sequel, Final Fantasy IV: The After Years is set seventeen years after Final Fantasy IV and includes most of the characters from the original game as well as introducing several new characters. The story of The After Years primarily revolves around Ceodore Harvey, the son of Cecil and Rosa. In 2011, a third game in the series was released. Set one year after Final Fantasy IV and sixteen years prior to The After Years, Final Fantasy IV Interlude, was released for the PlayStation Portable, and featured several of the original Final Fantasy IV protagonists.

Graph (discrete mathematics)

In mathematics, and more specifically in graph theory, a graph is a representation of a set of objects where some pairs of objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices (also called nodes or points), and the links that connect some pairs of vertices are called edges (also called arcs or lines). Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.

The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this is an undirected graph, because if person A shook hands with person B, then person B also shook hands with person A. In contrast, if there is an edge from person A to person B when person A knows of person B, then this graph is directed, because knowledge of someone is not necessarily a symmetric relation (that is, one person knowing another person does not necessarily imply the reverse; for example, many fans may know of a celebrity, but the celebrity is unlikely to know of all their fans). The former type of graph is called an undirected graph and the edges are called undirected edges while the latter type of graph is called a directed graph and the edges are called directed edges.

Edge (educational foundation)

Edge Foundation, commonly referred to as 'Edge', is an independent education foundation, dedicated to raising the status of practical and vocational learning in the UK. Its aim is for young people to have the opportunity to achieve their potential, to ensure that the UK’s future workforce is equipped with the skills to succeed.

Edge believes that there are many paths to success. They think that vocational and practical education should be valued just as highly as academic choices. To make this happen, they run campaigns and projects that challenge academic snobbery, stimulate demand for practical learning, and supply innovative learning opportunities.

Aims

As well as a better public perception of practical learning, Edge would like to see fundamental changes in the UK education system, including better careers guidance, more opportunities for practical and work-based learning, more employer engagement and an overhaul of teacher training, especially in practical subjects.